In the paper "Conditional validity of inductive conformal predictors" by V. Vovk, the author considers a condition of the form $$P_{X\times Y}(Y\in\Gamma(X)\mid X=x)\geq1-\alpha,$$ where $\alpha\in[0,1]$ and $\Gamma:\mathcal{X}\rightarrow P(\mathcal{Y})$ is a set predictor. He then goes on and proves a theorem that for reasonable spaces this means that $\Gamma$ needs to output the full space $\mathcal{Y}$ for almost all nonatoms $x\in\mathcal{X}$.
Intuitively (or from a data science perspective) this makes sense. Since $x$ is a nonatom, there is no way that any model can have enough information to accurately estimate these prediction regions. The only set satisfying the condition that can be constructed without much information is the whole space $\mathcal{Y}$.
However, for nonatoms, the conditional probability is not well-defined as far as I know. Since Vovk is usually a very meticulous and formal mathematician, I assume that there is a good reason that he can write this equation down unambiguously. But my question is then "how can we write this equation in a well-defined way"?